Sommersemester 2026

Stochastische Geometrie und räumliche Statistik

Vorträge:

_{14.04.     Christian Wolff – Deutsches Forschungszentrum für Künstliche Intelligenz (DFKI)}

 

Title: Collection, Processing and Evaluation of Human Gait Data Based on Ground Reaction Forces 
  

Abstract: Gait analysis is a cornerstone of understanding human locomotion and diagnosing musculoskeletal and neurological disorders. While gold-standard methods like optical motion capture and force plates deliver precise measurements, their reliance on controlled laboratory environments limits their practical use in everyday clinical settings, specifically trauma treatment. Wearable insoles, capable of continuously recording ground reaction forces (GRF) in real-world scenarios, offer new, transformative opportunities. Yet, their potential is constrained by the lack of standardized frameworks for data processing and analysis. This talk explores the challenges and opportunities in collecting, processing, and evaluating GRF-based gait data, with a focus on tibial fracture recovery and the development of robust analytical tools for clinical applications. We present a streamlined data processing workflow and introduce novel parameters for stance phase analysis, alongside a software solution designed to standardize and analyze GRF data from wearable devices. The talk will address key challenges, including the complexities of pathological gait patterns, hardware limitations, empirical data collection and the integration of data-driven approaches with machine learning. Through empirical studies, we examine the potential of GRF data to monitor rehabilitation progress and detect complications such as nonunion in tibial fracture patients. By bridging technological innovation with clinical practice, this work aims to enhance the precision and accessibility of gait analysis, ultimately improving patient outcomes and supporting evidence-based decision-making in both research and clinical care.

Dienstag, 14. April 2026, 16:00 Uhr, Helmholtzstr. 18, Raum 220

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_{12.05      Prof. Dr. Sergey Foss - (Heriot-Watt University, Edinburgh)}
 

Title: Barak-Erdos directed random graphs and related models: limit theorems, perfect simulation and around

Abstract: We consider directed random graphs, the prototype of which being the Barak-Erdős graph on the integers, and study the way that long (or heavy, if weights are present) paths grow. This is done by relating the graphs to certain particle systems that we call Infinite Bin Models (IBM).  We formulate a number of limit theorems. Regenerative techniques are used where possible, exhibiting random sets of vertices over which the graphs regenerate. When edges have random weights we show how the last passage percolation constants behave and when central limit theorems exist.
When the underlying vertex set is partially ordered, new phenomena occur, e.g., there are relations with last passage Brownian percolation. We also look at weights that may possibly take negative values and study in detail some special cases that require combinatorial/graph theoretic techniques that exhibit some interesting non-differentiability properties of the last passage percolation constant. We also explain how to approach the problem of estimation of last passage percolation constants by means of perfect simulation. The talk is based on a series of joint papers with Takis Konstantopoulos, Bastien Mallein, Sanjay Ramassami, James Martin and several other colleagues, and in particular on the survey paper.

https://arxiv.org/abs/2312.02884"

_{Dienstag, 12. Mai 2026, 16:00 Uhr, Helmholtzstr. 18, Raum 220}

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_{19.05.    Prof. Dr. Günter Last – KIT - Karlsruhe Institut für Technologie}

  

Title: Chaos expansion and Malliavin calculus for the Dirichlet–Ferguson process
 
Abstract: The Dirichlet–Ferguson process ζ is a random, purely discrete probability measure whose finite-dimensional distributions are Dirichlet distributions. It can be defined on a general state space and has numerous applications, such as in population genetics. We shall present the fundamental chaos expansion by Peccati (2008), providing an explicit formula for the kernel functions. We proceed with developing a Malliavin calculus for ζ. To this end, we introduce a gradient, divergence and a generator which act as linear operators on ζ-measurable random variables or random fields and which are linked by some basic formulas such as integration by parts. While this calculus is motivated by Malliavin calculus for isonormal Gaussian processes and the general Poisson process, the strong dependence properties of ζ require considerably more combinatorial efforts. We will identify our generator as the generator of the Fleming–Viot process and describe the associated Dirichlet form explicitly
in terms of the chaos expansion. If time permits, we shall also present a short direct proof of the Poincaré inequality.

The talk is based on joint work with Babette Picker (Karlsruhe).

_{Dienstag, 19. Mai 2026, 16:00 Uhr, Helmholtzstr. 18, Raum 220}

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_{23.06.    Prof. Dr. Rafik Aramyan - Institute of Mathematics der National Academy of Sciences of Armenia} 

Title: Inversion of the Two-Data Funk transform 
 

Abstract: It is known that the Funk transform (FT) is invertible in the class of even (symmetric) continuous functions defined on the unit 2-sphere . In this article, for the reconstruction of  a continuous function (can be non-even), an additional condition is found, which is a weighted Funk transform (to reconstruct an odd function), and the injectivity of the so-called two data Funk transform is considered. The transform consists of the classical FT and the weighted FT. An iterative inversion formula of the transform is presented. Such inversions have theoretical significance in convexity theory, integral geometry and spherical tomography. 

_{Dienstag, 23. Juni, 2026, 16:00 Uhr, Helmholtzstr. 18, Raum 220}

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_{14.07.     Dr. Mikhail Temkin - Universität Freiburg}

 

Title: How to geometrize and strengthen interlevel persistence

Abstract: For a function f on a topological space X, Cohen-Steiner--Edelsbrunner--Harer introduced the notion of extended barcode of f. It is a combinatorial gadget that compactly stores information about, in particular, homology of any interlevel set f^{-1}([a,b]) (over a field F). When X is a closed manifold and f is Morse without multiple critical values, we propose an equivalent, geometrical definition of extended barcode. It relies on a Bruhat decomposition, which describes the relative position of two complete flags in a vector space. As an important byproduct of this approach we naturally associate a number (i.e. an element of F) to every bar. It turns out that in certain cases the product of all the numbers doesn't depend on the function. In one such case, this product equals the Reidemeister torsion of X. Time permitting, we will also discuss the behavior of these objects in families of functions. Based on several works, one is joint with P. Pushk.
 

_{Dienstag, 14. Juli, 2026, 16:00 Uhr, Helmholtzstr. 18, Raum 220}

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_{04.08.     Manuel Alfano - Politecnico di Milano}
 

Title: Adaptive Delayed-Acceptance MCMC with Gaussian-Process  Surrogates for Bayesian Inverse Problems: From Active Learning to Ergodicity

Abstract: Bayesian inverse problems often require a large number of evaluations of an expensive high-fidelity forward model, making Standard Markov chain Monte Carlo sampling computationally demanding. This work develops two closely related adaptive multi-fidelity algorithms that combine delayed acceptance, Gaussian-process surrogates, and active learning. The first algorithm is the original computational formulation. Along the MCMC path, the surrogate is used when its predictive uncertainty is below a prescribed threshold; otherwise, the high-fidelity model is evaluated and the new information is used to update the surrogate online. The frequency of high-fidelity corrections can also be adapted according to the discrepancy between low-and high-fidelity outputs. This is the most flexible and practically motivated scheme, but a complete proof of ergodicity is not established. To obtain a rigorous convergence  result, a theoretically tractable variant is then introduced. The Gaussian-process surrogate is frozen during each macro-iteration and defines an approximate posterior together with an m-step coarse Metropolis-Hastings proposal. The proposal is corrected by a delayed-acceptance step based on the exact model, and the surrogate is enriched only after this  correction using coarse-chain points whose predictive uncertainty exceeds a decreasing threshold. Under compactness, positivity, fixed-kernel noise-free GP, regularity, and full-support assumptions, each frozen kernel is reversible with respect to the exact posterior; simultaneous containment and  diminishing adaptation are also established. Consequently, the adaptive chain converges to the exact posterior in total variation and satisfies a weak law of large numbers. The two algorithms therefore provide a practical active-learning strategy and a rigorous theoretical counterpart for surrogate-accelerated Bayesianinversion.

 

Keywords: Bayesian inverse problems; adaptive MCMC; delayed acceptance; Gaussian processes; active learning; ergodicity. 

_{Dienstag, 04. August, 2026, 16:00 Uhr, Helmholtzstr. 18, Raum 220}

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_{11.08.    Dominik Pabst - Friedrich-Alexander-Universität Elangen-Nürnberg}
 

Title: Analysis of Betti Numbers and Estimation of Minkowski Tensors: Theory and Algorithms in Stochastic Geometry
 

Abstract: Random spatial structures arise naturally in mathematics, statistics, physics and materials science. This talk highlights two research projects that illustrate different mathematical approaches to their analysis. The first part concerns a new model of random simplicial complexes based on the random connection model. In comparison to graphs, simplicial complexes also allow the modeling of interactions between more than two objects that go beyond pairwise interactions. Within this model, a central limit theorem is established for a broad class of functionals, including the Betti numbers describing the topological structure of an object. Moreover, the model can be constructed in such a way that the resulting simplicial complex has the same topological properties as the Boolean model. This allows the central limit theorem to be transferred directly to the Betti numbers of the Boolean model. The second part focuses on the estimation of Minkowski tensors. Minkowski tensors are tensor-valued generalizations of the classical Minkowski functionals which, in two dimensions, consist of area, boundary length and Euler characteristic. While the Minkowski functionals provide scalar geometric descriptors, Minkowski tensors additionally capture directional information such as shape,  orientation and anisotropy. An asymptotically unbiased estimation algorithm based solely on point samples is presented. The corresponding open-source implementation is applicable in arbitrary dimensions and for tensors of arbitrary rank. The resulting method provides a practical tool for the quantitative analysis of complex spatial structures arising in applications. 

_{Dienstag, 11. August 2026, 16:00 Uhr, Helmholtzstr. 18, Raum 220}

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_{18.08.     Georgia Lupton -  Liverpool University - UK}
 

Title: Analytical Reconstruction of Anisotropic Power Diagrams

Abstract: Typically, when finding the boundaries of an Anisotropic Power diagram (also known as Generalised Balanced Power Diagram) methods evaluate the points on the boundary of each cell which give a discretised version of the boundaries. Deriving analytical boundaries can be challenging. A recently introduced method for computing the analytical boundary of 2D anisotropic power diagrams has a time complexity of O(n^4), where n represents the number of cells. While this approach successfully determines the boundaries, its high computational cost makes it inefficient for large cell counts. Therefore, in my talk, I will discuss a new algorithm which implements a sweep circle to reduce the time taken for large number of cells and is adaptable for other tessellations and metrics. I will also briefly cover some aspects of discrete tomography that has been a large part of my PhD research, specifically U-polygons (where a convex polygon P is a U-polygon for a finite set U of directions, if every vertex v of P has a line v +lu that pass through another vertex v′ for all u in U), when they can be embedded in the integer lattice and how this links to the task of reconstructing convex sets from tomographic data.
 

_{Dienstag, 18. August, 2026, 16:00 Uhr, Helmholtzstr. 18, Raum 220}

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_Gäste:

• Christian Wolff – Deutsches Forschungszentrum für Künstliche Intelligenz (DFKI)

• Prof. Dr. Sergey Foss - (Heriot-Watt University, Edinburgh)

• Prof. Dr. Günter Last – KIT - Karlsruhe Institut für Technologie

• Prof. Dr. Rafik Aramyan - Institute of Mathematics der National Academy of Sciences of Armenia 

• Dr. Mikhail Temkin - Universität Freiburg

• Manuel Alfano - Politecnico di Milano

• Dominik Pabst - Friedrich-Alexander-Universität Elangen-Nürnberg

• Georgia Lupton -  Liverpool University - UK